doi: 10.3934/dcdss.2020419

Global weak solutions for an newtonian fluid interacting with a Koiter type shell under natural boundary conditions

1. 

Bahnhofstr. 39, 71364 Winnenden, Germany

2. 

Mathematisches Institut, Ernst-Zermelo-Str. 1, 79104 Freiburg, Germany

* Corresponding author: Michael Růžička

Received  January 2020 Revised  June 2020 Published  August 2020

We consider a viscous, incompressible Newtonian fluid flowing through a thin elastic (non-cylindrical) structure. The motion of the structure is described by the equations of a linearised Koiter shell, whose motion is restricted to transverse displacements. The fluid and the structure are coupled by the continuity of velocities and an equilibrium of surface forces on the interface between fluid and structure. On a fixed in- and outflow region we prescribe natural boundary conditions. We show that weak solutions exist as long as the shell does not self-intersect.

Citation: Hannes Eberlein, Michael Růžička. Global weak solutions for an newtonian fluid interacting with a Koiter type shell under natural boundary conditions. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020419
References:
[1]

G. AcostaR. G. Durán and F. López García, Korn inequality and divergence operator: counterexamples and optimality of weighted estimates, Proc. Amer. Math. Soc., 141 (2013), 217-232.  doi: 10.1090/S0002-9939-2012-11408-X.  Google Scholar

[2]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis, 3rd edition, Springer, Berlin, 2006, A hitchhiker's guide.  Google Scholar

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H. Beirão da Veiga, On the existence of strong solutions to a coupled fluid-structure evolution problem, J. Math. Fluid Mech., 6 (2004), 21-52.  doi: 10.1007/s00021-003-0082-5.  Google Scholar

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J.-M. E. Bernard, Density results in Sobolev spaces whose elements vanish on a part of the boundary, Chin. Ann. Math. Ser. B, 32 (2011), 823-846.  doi: 10.1007/s11401-011-0682-z.  Google Scholar

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F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183 of Applied Mathematical Sciences, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

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R. M. Brown and Z. Shen, Estimates for the Stokes operator in Lipschitz domains, Indiana Univ. Math. J., 44 (1995), 1183-1206.  doi: 10.1512/iumj.1995.44.2025.  Google Scholar

[8]

A. ChambolleB. DesjardinsM. J. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech., 7 (2005), 368-404.  doi: 10.1007/s00021-004-0121-y.  Google Scholar

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P. G. Ciarlet, Mathematical Elasticity. Vol. Ⅲ, vol. 29 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 2000  Google Scholar

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P. G. Ciarlet, An Introduction to Differential Geometry with Applications to Elasticity, With a foreword by Roger Fosdick. J. Elasticity 78/79 no. 1-3 (2005). doi: 10.1007/s10659-005-4738-8.  Google Scholar

[11]

G. Cokelet, The rheology and tube flow of blood, in Handbook of bioengineering (eds. R. Skalak and S. Chien), McGraw-Hill, New York, 1987. Google Scholar

[12]

D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 303-352.  doi: 10.1007/s00205-005-0385-2.  Google Scholar

[13]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[14]

M. Dobrowolski, Angewandte Funktionalanalysis: Funktionalanalysis, Sobolev-Räume und Elliptische Differentialgleichungen, Springer-Lehrbuch Masterclass, Springer-Verlag, Berlin, 2006. Google Scholar

[15]

J. Droniou, Intégration et Espaces de Sobolev à Valeurs Vectorielles., 2001, URL https://hal.archives-ouvertes.fr/hal-01382368. Google Scholar

[16]

H. Eberlein, Globale Existenz Schwacher Lösungen für die Interaktion Eines Newtonschen Fluides mit Einer linearen, Transversalen Koiter-Schale unter Natürlichen Randbedingungen, PhD thesis, Albert-Ludwigs-Universität Freiburg im Breisgau, 2017. Google Scholar

[17]

E. B. FabesC. E. Kenig and G. C. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J., 57 (1988), 769-793.  doi: 10.1215/S0012-7094-88-05734-1.  Google Scholar

[18]

C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal., 40 (2008), 716-737.  doi: 10.1137/070699196.  Google Scholar

[19]

E.-i. Hanzawa, Classical solutions of the Stefan problem, Tôhoku Math. J., 33 (1981), 297-335.  doi: 10.2748/tmj/1178229399.  Google Scholar

[20]

T. KatoM. MitreaG. Ponce and M. Taylor, Extension and representation of divergence-free vector fields on bounded domains, Math. Res. Lett., 7 (2000), 643-650.  doi: 10.4310/MRL.2000.v7.n5.a10.  Google Scholar

[21]

V. Lakshmikantham and M. Rama Mohana Rao, Theory of Integro-Differential Equations, vol. 1 of Stability and Control: Theory, Methods and Applications, Gordon and Breach Science Publishers, Lausanne, 1995.  Google Scholar

[22]

D. Lengeler, Globale Existenz für Die Interaktion Eines Navier-Stokes-Fluids Mit Einer Linear Elastischen Schale, PhD thesis, Albert-Ludwigs-Universität Freiburg im Breisgau, 2011, Urn: nbn: de: bsz: 25-opus-84219. Google Scholar

[23]

D. Lengeler, Weak solutions for an incompressible, generalized Newtonian fluid interacting with a linearly elastic Koiter type shell, SIAM J. Math. Anal., 46 (2014), 2614-2649.  doi: 10.1137/130911299.  Google Scholar

[24]

D. Lengeler and M. Růžička, Weak solutions for an incompressible Newtonian fluid interacting with a Koiter type shell, Arch. Ration. Mech. Anal., 211 (2014), 205-255.  doi: 10.1007/s00205-013-0686-9.  Google Scholar

[25]

J. Lequeurre, Existence of strong solutions for a system coupling the Navier-Stokes equations and a damped wave equation, J. Math. Fluid Mech., 15 (2013), 249-271.  doi: 10.1007/s00021-012-0107-0.  Google Scholar

[26]

J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[27]

J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and Measure-Valued Solutions to Evolutionary PDEs, vol. 13 of Applied Mathematics and Mathematical Computation, Chapman & Hall, London, 1996.  Google Scholar

[28]

J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Dover Publications, Inc., New York, 1994, Corrected reprint of the 1983 original.  Google Scholar

[29]

N. Masmoudi and F. Rousset, Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equations, Arch. Ration. Mech. Anal., 223 (2017), 301–417, URL https://doi.org/10.1007/s00205-016-1036-5. doi: 10.1007/s00205-016-1036-5.  Google Scholar

[30]

B. Muha and S. Čanić, Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls, Arch. Ration. Mech. Anal., 207 (2013), 919-968.  doi: 10.1007/s00205-012-0585-5.  Google Scholar

[31]

B. Muha and S. Čanić, A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof, Commun. Inf. Syst., 13 (2013), 357-397.  doi: 10.4310/CIS.2013.v13.n3.a4.  Google Scholar

[32]

B. Muha and S. Čanić, Fluid-structure interaction between an incompressible, viscous 3D fluid and an elastic shell with nonlinear Koiter membrane energy, Interfaces Free Bound., 17 (2015), 465-495.  doi: 10.4171/IFB/350.  Google Scholar

[33]

B. Muha and S. Čanić, Existence of a weak solution to a fluid-elastic structure interaction problem with the Navier slip boundary condition, J. Differential Equations, 260 (2016), 8550-8589.  doi: 10.1016/j.jde.2016.02.029.  Google Scholar

[34]

A. QuarteroniM. Tuveri and A. Veneziani, Computational vascular fluid dynamics: problems, models and methods, Computing and Visualization in Science, 2 (2000), 163-197.  doi: 10.1007/s007910050039.  Google Scholar

[35]

R. Russo, On Stokes' problem, in Advances in Mathematical Fluid Mechanics, Springer, Berlin, 2010,473–511. doi: 10.1007/978-3-642-04068-9_28.  Google Scholar

[36]

Z. W. Shen, A note on the Dirichlet problem for the Stokes system in Lipschitz domains, Proc. Amer. Math. Soc., 123 (1995), 801-811.  doi: 10.1090/S0002-9939-1995-1223521-9.  Google Scholar

[37]

H. Sohr, The Navier-Stokes Equations, An elementary functional analytic approach. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8255-2.  Google Scholar

[38]

M. E. Taylor, Partial Differential Equations I. Basic theory, vol. 115 of Applied Mathematical Sciences, Second edition edition, Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8.  Google Scholar

[39]

R. Temam, Navier-Stokes Equations. Theory and numerical analysis, North-Holland Publishing Co., Amsterdam-New York, 1979, Studies in Mathematics and its Applications, Vol. 2.  Google Scholar

show all references

References:
[1]

G. AcostaR. G. Durán and F. López García, Korn inequality and divergence operator: counterexamples and optimality of weighted estimates, Proc. Amer. Math. Soc., 141 (2013), 217-232.  doi: 10.1090/S0002-9939-2012-11408-X.  Google Scholar

[2]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis, 3rd edition, Springer, Berlin, 2006, A hitchhiker's guide.  Google Scholar

[3]

H. Beirão da Veiga, On the existence of strong solutions to a coupled fluid-structure evolution problem, J. Math. Fluid Mech., 6 (2004), 21-52.  doi: 10.1007/s00021-003-0082-5.  Google Scholar

[4]

J.-M. E. Bernard, Density results in Sobolev spaces whose elements vanish on a part of the boundary, Chin. Ann. Math. Ser. B, 32 (2011), 823-846.  doi: 10.1007/s11401-011-0682-z.  Google Scholar

[5]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183 of Applied Mathematical Sciences, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[6]

H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert, vol. 5 of North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 1973. Google Scholar

[7]

R. M. Brown and Z. Shen, Estimates for the Stokes operator in Lipschitz domains, Indiana Univ. Math. J., 44 (1995), 1183-1206.  doi: 10.1512/iumj.1995.44.2025.  Google Scholar

[8]

A. ChambolleB. DesjardinsM. J. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech., 7 (2005), 368-404.  doi: 10.1007/s00021-004-0121-y.  Google Scholar

[9]

P. G. Ciarlet, Mathematical Elasticity. Vol. Ⅲ, vol. 29 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 2000  Google Scholar

[10]

P. G. Ciarlet, An Introduction to Differential Geometry with Applications to Elasticity, With a foreword by Roger Fosdick. J. Elasticity 78/79 no. 1-3 (2005). doi: 10.1007/s10659-005-4738-8.  Google Scholar

[11]

G. Cokelet, The rheology and tube flow of blood, in Handbook of bioengineering (eds. R. Skalak and S. Chien), McGraw-Hill, New York, 1987. Google Scholar

[12]

D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 303-352.  doi: 10.1007/s00205-005-0385-2.  Google Scholar

[13]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[14]

M. Dobrowolski, Angewandte Funktionalanalysis: Funktionalanalysis, Sobolev-Räume und Elliptische Differentialgleichungen, Springer-Lehrbuch Masterclass, Springer-Verlag, Berlin, 2006. Google Scholar

[15]

J. Droniou, Intégration et Espaces de Sobolev à Valeurs Vectorielles., 2001, URL https://hal.archives-ouvertes.fr/hal-01382368. Google Scholar

[16]

H. Eberlein, Globale Existenz Schwacher Lösungen für die Interaktion Eines Newtonschen Fluides mit Einer linearen, Transversalen Koiter-Schale unter Natürlichen Randbedingungen, PhD thesis, Albert-Ludwigs-Universität Freiburg im Breisgau, 2017. Google Scholar

[17]

E. B. FabesC. E. Kenig and G. C. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J., 57 (1988), 769-793.  doi: 10.1215/S0012-7094-88-05734-1.  Google Scholar

[18]

C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal., 40 (2008), 716-737.  doi: 10.1137/070699196.  Google Scholar

[19]

E.-i. Hanzawa, Classical solutions of the Stefan problem, Tôhoku Math. J., 33 (1981), 297-335.  doi: 10.2748/tmj/1178229399.  Google Scholar

[20]

T. KatoM. MitreaG. Ponce and M. Taylor, Extension and representation of divergence-free vector fields on bounded domains, Math. Res. Lett., 7 (2000), 643-650.  doi: 10.4310/MRL.2000.v7.n5.a10.  Google Scholar

[21]

V. Lakshmikantham and M. Rama Mohana Rao, Theory of Integro-Differential Equations, vol. 1 of Stability and Control: Theory, Methods and Applications, Gordon and Breach Science Publishers, Lausanne, 1995.  Google Scholar

[22]

D. Lengeler, Globale Existenz für Die Interaktion Eines Navier-Stokes-Fluids Mit Einer Linear Elastischen Schale, PhD thesis, Albert-Ludwigs-Universität Freiburg im Breisgau, 2011, Urn: nbn: de: bsz: 25-opus-84219. Google Scholar

[23]

D. Lengeler, Weak solutions for an incompressible, generalized Newtonian fluid interacting with a linearly elastic Koiter type shell, SIAM J. Math. Anal., 46 (2014), 2614-2649.  doi: 10.1137/130911299.  Google Scholar

[24]

D. Lengeler and M. Růžička, Weak solutions for an incompressible Newtonian fluid interacting with a Koiter type shell, Arch. Ration. Mech. Anal., 211 (2014), 205-255.  doi: 10.1007/s00205-013-0686-9.  Google Scholar

[25]

J. Lequeurre, Existence of strong solutions for a system coupling the Navier-Stokes equations and a damped wave equation, J. Math. Fluid Mech., 15 (2013), 249-271.  doi: 10.1007/s00021-012-0107-0.  Google Scholar

[26]

J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[27]

J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and Measure-Valued Solutions to Evolutionary PDEs, vol. 13 of Applied Mathematics and Mathematical Computation, Chapman & Hall, London, 1996.  Google Scholar

[28]

J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Dover Publications, Inc., New York, 1994, Corrected reprint of the 1983 original.  Google Scholar

[29]

N. Masmoudi and F. Rousset, Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equations, Arch. Ration. Mech. Anal., 223 (2017), 301–417, URL https://doi.org/10.1007/s00205-016-1036-5. doi: 10.1007/s00205-016-1036-5.  Google Scholar

[30]

B. Muha and S. Čanić, Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls, Arch. Ration. Mech. Anal., 207 (2013), 919-968.  doi: 10.1007/s00205-012-0585-5.  Google Scholar

[31]

B. Muha and S. Čanić, A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof, Commun. Inf. Syst., 13 (2013), 357-397.  doi: 10.4310/CIS.2013.v13.n3.a4.  Google Scholar

[32]

B. Muha and S. Čanić, Fluid-structure interaction between an incompressible, viscous 3D fluid and an elastic shell with nonlinear Koiter membrane energy, Interfaces Free Bound., 17 (2015), 465-495.  doi: 10.4171/IFB/350.  Google Scholar

[33]

B. Muha and S. Čanić, Existence of a weak solution to a fluid-elastic structure interaction problem with the Navier slip boundary condition, J. Differential Equations, 260 (2016), 8550-8589.  doi: 10.1016/j.jde.2016.02.029.  Google Scholar

[34]

A. QuarteroniM. Tuveri and A. Veneziani, Computational vascular fluid dynamics: problems, models and methods, Computing and Visualization in Science, 2 (2000), 163-197.  doi: 10.1007/s007910050039.  Google Scholar

[35]

R. Russo, On Stokes' problem, in Advances in Mathematical Fluid Mechanics, Springer, Berlin, 2010,473–511. doi: 10.1007/978-3-642-04068-9_28.  Google Scholar

[36]

Z. W. Shen, A note on the Dirichlet problem for the Stokes system in Lipschitz domains, Proc. Amer. Math. Soc., 123 (1995), 801-811.  doi: 10.1090/S0002-9939-1995-1223521-9.  Google Scholar

[37]

H. Sohr, The Navier-Stokes Equations, An elementary functional analytic approach. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8255-2.  Google Scholar

[38]

M. E. Taylor, Partial Differential Equations I. Basic theory, vol. 115 of Applied Mathematical Sciences, Second edition edition, Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8.  Google Scholar

[39]

R. Temam, Navier-Stokes Equations. Theory and numerical analysis, North-Holland Publishing Co., Amsterdam-New York, 1979, Studies in Mathematics and its Applications, Vol. 2.  Google Scholar

Figure 1.  Reference domain $ \Omega $ with in- and outflow region $ \Gamma $ and moving boundary $ M $
Figure 2.  Notations for admissible in- and outflow domains and moving domains
Figure 3.  Extension of the fluid domain
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